Measuring tropical rainforest resilience under non-Gaussian disturbances

The Amazon rainforest is considered one of the Earth’s tipping elements and may lose stability under ongoing climate change. Recently a decrease in tropical rainforest resilience has been identified globally from remotely sensed vegetation data. However, the underlying theory assumes a Gaussian distribution of forest disturbances, which is different from most observed forest stressors such as fires, deforestation, or windthrow. Those stressors often occur in power-law-like distributions and can be approximated by α-stable Lévy noise. Here, we show that classical critical slowing down (CSD) indicators to measure changes in forest resilience are robust under such power-law disturbances. To assess the robustness of CSD indicators, we simulate pulse-like perturbations in an adapted and conceptual model of a tropical rainforest. We find few missed early warnings and few false alarms are achievable simultaneously if the following steps are carried out carefully: first, the model must be known to resolve the timescales of the perturbation. Second, perturbations need to be filtered according to their absolute temporal autocorrelation. Third, CSD has to be assessed using the non-parametric Kendall-τ slope. These prerequisites allow for an increase in the sensitivity of early warning signals. Hence, our findings imply improved reliability of the interpretation of empirically estimated rainforest resilience through CSD indicators.


Introduction
The Amazon rainforest is considered a crucial component of the Earth's climate system [1] and has been suggested as an Earth system tipping element [2][3][4].There is growing concern that various anthropogenic stressors, such as climate change and associated changes in rainfall patterns, fires, land-use change and deforestation, cause a decrease in resilience and could ultimately lead to large-scale shifts in the Amazons' ecosystem, with severe consequences for the biosphere and human societies [5][6][7][8].Based on conceptual models and observational data, it is believed that the rainforest exhibits the potential for multi-stability at specific levels of moisture supply [9][10][11][12][13].This means that certain regions of the rainforest may transition from a rainforest to a savanna-like vegetation state if local precipitation rates are decreased below critical thresholds.
Tropical rainforests, such as the Amazon basin, are subject to multiple stressors.These can originate due to climate (e.g.droughts, heat waves, windthrows), hydrology (e.g.landslides), biotic factors (e.g.insect outbreaks) or anthropogenic activity (e.g.deforestation, wildfires).Many such stressors cause disturbance events, which in turn change the forests in a pulse-like manner.The consequences of disturbances are often visible in the canopy gap structure, i.e. single disturbance events destroy destroy entire parts of the forest, while neighboring parts are almost completely undisturbed.This gap structure has been observed to follow power-law-like distributions [14][15][16][17][18][19][20][21][22].In other words, there is a scale-free nature to the gaps, and very large gaps are likely.These power-law-like distributions have also been observed directly for both droughts [23] and wildfires [24] in different ecoregions worldwide.Hence, if we understand disturbances as random perturbations to the forest, i.e. as noise, this noise might have non-Gaussian characteristics.For instance, the noise could be heavy-tailed (e.g.powerlaw tails).Thus, extreme events become more likely than under Gaussian white noise.Furthermore, Gaussian noise would lead to continuous forest state evolutions, whereas noise due to disturbance events would result in discrete jumps in the time series.Of course, describing all disturbances by one abstract probability distribution is a highly reductive approach given the complexity of the system, yet if it were to be done, a non-Gaussian distribution should not be ruled out.
The resilience of tropical rainforests is usually measured as the response to disturbances directly from observed time series data.The recovery rate to disturbances is related to the temporal autocorrelation of a time series.If a forest is resilient and quickly recovers, the autocorrelation is lower compared to a higher autocorrelation of a vulnerable forest that slowly recovers.Increased autocorrelation in tropical rainforests worldwide has been found using different remotely-sensed vegetation proxies such as above-ground biomass (accounting for water stress, deforestation and vapor-pressure deficit, [25]), vegetation optical-depth ( [7] only Amazon basin, and [26] globally), normalized difference vegetation index (NDVI) [27] and kernel NDVI [28].
Interpreting the autocorrelation as an indicator of resilience requires certain mathematical assumptions [29,30], under which the autocorrelation can also serve as an early warning signal in the approach to a critical transition [31].The underlying phenomenon, critical slowing down (CSD), traces a decreasing ability of a system to recover from perturbations when it loses stability.Typically, Gaussian noise is assumed.This is because an analytical relationship between the classical CSD indicators variance and temporal autocorrelation of lag one and the recovery rate λ of the dynamics linearized around a given equilibrium can be established [32,33] (see below).However, as it is not clear that the disturbances occurring in tropical rainforests can be described with Gaussian noise, it is also not clear if the forest resilience can be measured via the autocorrelation.
In this paper, we study forest resilience indicators in the non-Gaussian noise case.For this, we consider α-stable Lévy noise, a heavy-tailed generalization of the Gaussian distribution.Depending on the parameters chosen, the noise time series contains jumps (i.e.discrete disturbances events) and the tails follow a power-law (i.e.extreme events become more likely).These characteristics are similar to the ones we postulate for disturbances in tropical rainforests.Hence, it is necessary to understand whether the CSD-based resilience indicators still perform well for Lévy noise.Additionally, we look at pink noise, which contains power-law tails, but no jumps.Previous work has already looked at red noise and time-correlated noise [32][33][34].Of course, our approach of employing 1D non-Gaussian noise distributions is reductive, as it can not represent any temporal and spatial patterns [35,36].Yet, since the presence of non-Gaussian noise is plausible for tropical rainforests, the correct functioning of resilience indicators and early warning signals before critical transitions should be assessed thoroughly.
For this assessment, we use a simple conceptual tipping element model of a tropical rainforest [37].This model can represent a real tropical rainforest at the highest level of abstraction, capturing only the stabilising and destabilising feedbacks of the spatially extended system.It gives a good first-order approximation to the tipping structure [37,38], i.e. the bi-stability of tropical rainforests between rainforest and savannah states [9].For our analysis of forest resilience, it therefore constitutes a good starting point.The argument is a similar one as before: if the indicators do not work in such a simple setting, it is unlikely they would work if real rainforests were to be observed or modelled in more detail.(CSD), such as the autocorrelation.The phenomenon describes a decreased stability of a system up to the point where the system undergoes a critical transition and switches into another state.Hence, the estimation of forest resilience is typically an identical process as the study of early warning to critical transition, such as early warning of tipping elements in the Earth system.In order to test the reliability of early warning signals (forest resilience indicators), we simulate rainforest-savannah transitions of a tropical rainforest using the conceptual model by van Nes et al. [37].Thus, our theoretical approach provides evidence necessary to understand actual measurements that may be influenced by non-Gaussian noise.

Conceptual
In this model, the tree cover T [%] of the rainforest is a tri-stable system with forest, savannah and treeless states depending on the precipitation P [mm day −1 ] as an external forcing.Ignoring the treeless state, the model dynamics locally exhibit the following characteristics: for 2 mm day −1 < P < 2.94 mm day −1 only a stable savannah state exists, for 2.95 mm day −1 < P < 4.41 mm day −1 both a stable savannah and a stable forest state exist and for 4.42 mm day −1 < P < 5 mm day −1 only a stable forest state exists (Fig. 1a).These regimes result from a governing equation [37], which without displaying units reads Here, the first term describes a logistic growth of tree cover with a precipitationdependent expansion coefficient.The second term accounts for the Allee effect: if the tree cover is low, new trees have a harder time growing because they have less protective covering from older trees.The third term introduces a wildfire effect: dense forests are subject to higher fire mortality.While this model represents a strongly stylised way to model the dynamics of tropical vegetation, its simplicity is central to our study as it allows for many simulations of critical transitions.
Model modification to include forest response on small time scales.Originally, the van Nes et al. [37] tipping model was developed to model critical transitions on long time scales.To investigate early warning for such critical transitions, a system response on short time scales is necessary because all critical slowing down indicators are based on an increasing disturbance recovery time as a tipping point is approached.In the original van Nes et al. [37] rainforest model, recovery from even very small disturbances would take hundreds of years (Fig. 2b, dashed lines).This can be loosely understood as the removal of a few trees and subsequent regrowth.However, small changes in the rainforest state T could also be understood as a response to climate variations, i.e. small fluctuations due to weather.For instance, during a drier than usual period, trees might develop fewer leaves, but already a couple of months later they could completely recover (compare [39][40][41][42]).Note that the forest response to weather can be highly nonlinear, as trees have several regulating mechanisms (e.g.stomatal control or hydraulic resistance), which may render the forest fairly stable even under strong climatic fluctuations.Still, it is reasonable to argue that there is the possibility for the intact rainforest to quickly recover from small perturbations which may loosely be understood as climate variations.
We introduce this net response to small fluctuations into the model by adding an additional short-term resilience term to Eq. 1.This leads to the following changes in the potential: around the stable fixed point, we create an additional potential valley, decreasing in depth as the rainforest-savannah tipping point is approached.Fig. 2a displays the changes: a little peak parametrized by a Gaussian exponential function centred at the rainforest attractor (right-most minimum of the potential) is subtracted from the original potential (Fig. 2a, dashed lines).This additional stabilizing force strongly reduces the return times from small disturbances (Fig. 2b, solid lines).The updated model equations, again displayed without units, are: Here, ∆T f ix (T, P ) = T − T f ix (P ) is the distance to the rainforest fixed point at given P .We chose the height and width of the Gaussian such that the return time of a 1% tree cover disturbance at P = 5 mm day −1 is approximately one month.The width is decreased exponentially (exp(5 − P )) with lower P and the height is decreased by a scaled sigmoid (tanh(25P − 75) + 1) such that close to the savannah transition it vanishes.In addition, the new term dN t represents noise increments.
Simulating rainforest disturbances with α-stable Lévy noise Multiple stressors influence tropical rainforests.Many of these occur as single, discrete, events.For instance, wildfires or windthrows may destroy parts of the forest within a few hours.The outcome is subsequently visible in the canopy gap structure.Commonly, the observed fragmentation patterns of tropical rainforests, in particular the Amazon, follows a power-law-like distribution [14-22, 43, 44].This means, patches of all sizes exist, and particularly large gaps are observed more frequently than they would be if their origin was a random Gaussian noise process.In reality, the disturbance distribution alone does not cause the observed gap structure.Instead, complex spatial and temporal mechanisms are at play.However, as droughts [23] and wildfires [24] can also follow power-law-like distributions, it is possible that random perturbations in tropical rainforests follow a non-Gaussian distribution, possibly with power-law tails.
In this work, we assess via simulations whether critical slowing down can be detected if the underlying noise distribution is not Gaussian but has power-law tails with jumps.For random variables with power-law tails (and thus infinite variance), a generalized central limit theorem holds [45,in §35 Theorem 5].The limit distributions belong to the family of α-stable Lévy noise [46].This is a family of distributions with parameters α ∈ (0, 2], β ∈ [−1, 1] that generalizes the Gaussian distribution (containing it at α = 2, β = 0).We choose β = −1, to simulate negative disturbances, the resulting tail behaviour in a log-log plot is shown in Fig. 1b.For α < 2, the tails follow a powerlaw [47,48].In this work, we use α-stable Lévy noise L α (σ; β) with amplitude σ as noise increments in our models: Such Lévy noise leads to jumps in the forest trajectory if α < 1, which could represent single, rapid events such as wildfires [24] or windthrows [44].
Early warning of regime shift with critical slowing down indicators When a system approaches a tipping point, critical slowing down measures the gradually declining recovery rate of the linearized dynamics and can be used as an early warning indicator [31,[49][50][51].Here, critical slowing down refers to an increase in the recovery time from perturbations as the system approaches a bifurcation.Measured by the rate of recovery from small perturbations, the phenomenon is used to assess ecological resilience and to warn before a critical transition is reached.The theoretical justification for using the standard deviation and the autocorrelation as critical slowing down indicators arises from first linearizing the dynamics around a given stable fixed point x * to obtain the linear Langevin equation where B t is a Wiener process.The solution is an Ornstein-Uhlenbeck process, for which analytic expressions of both classical CSD indicators can be derived [52]: where ∆t > 0 is the sampling time step.Clearly, in approach to a critical point at which the linear restoring rate vanishes (λ ↗ 0), both indicators increase monotonically . This statement holds also for Eq. 2 with a more general Gaussian noise term dN t .However, if the noise term follows an α-stable Lévy distribution with α < 2, variance and AC(1) are ill-defined because the second moment of x would be infinite.Since the EWS assessments inherent to this study are always performed on a bounded state space, this poses no practical problem and an analogous CSD characteristic to that seen for the Gaussian white noise case can indeed be analytically motivated (see Appendix C for further discussion).We will assess the numerical behaviour of the above indicators using simulations with disturbances that follow such power-law noise with jumps.Furthermore, we assess the performance of the interquartile range (IQR), an indicator of the width of a distribution similar to the standard deviation, yet more robust to outliers.Even though we do not give exact analytical expressions for the IQR, critical slowing down suggests that as a critical point is approached, the IQR increases monotonically (IQR ↗ ∞).
The concrete protocol follows Boers [52] and is depicted in Fig. 1c-d.We simulate 1000 model years of steady state, followed by 500 years of climate change and 500 years in a new steady state.During the simulated climate change, the forcing parameter precipitation is changed from 5.0 mm day −1 to 2.5 mm day −1 .Hence, the forest undergoes a critical transition with the rainforest state ceasing to exist around 1400 years at a value of 2.943 mm day −1 .We simulate realizations of systems under such forcing with sampled noise.To study the actual noise independent of the underlying drift, we subtract a deterministic trajectory from each stochastic realization.Note that this optimal way of nonlinearly detrending the time series -a necessary processing step prior to computing CSD indicators -is only possible in model systems.When working with observational data, one has to rely on suitable low-pass filtering.We then compute moving-window early warning signals with a window size of 10 model years.From these indicators, we compute a slope.We assess if the slope is increasing by comparing it with a null distribution of slopes of the same time series perturbed under random phase shifts.Early warning is then predicted if the observed slope is among the 5% highest slopes derived from the surrogate time series.

Results
Early warning with few misses and few false alarms is possible under Lévy noise according to our experiments.We quantify the rate of achieving accurate early warnings with the recall, the fraction of correctly predicted transitions (true positives, T P ) over all transitions (true positives and false negatives, F N ).In other terms, the recall is high if no critical transition is missed, and low, if there are too few early warnings.

Recall = T P T P + F N
To study false alarms, we assess our early warning classification pipeline over random 50-year windows of the simulated steady-state years (which have no critical transition).We quantify the rate of false alarms with the false positive rate (FPR), i.e. the fraction of transitions predicted (false positives, F P ) over all assessed transition-free time windows (false positives and true negatives, T N ).In other terms, the FPR is low if there are no false alarms, and high, if there are too many early warnings.
When resolving small timescales and assessing critical slowing down with the nonparametric Kendall-τ slope, we find an high average recall of 0.99 (fig.3a The robustness becomes clear in the third block, there is little variation in recall when comparing other early warning indicators than the interquartile-range (IQR).The final row shows that adjusting the early warning based on the absolute autocorrelation reduces the FPR by a factor of five, while keeping a high recall.
that is able to detect destabilization, i.e. a decline in the system's resilience.More specifically, the early warning pipeline displays a high recall (on average 0.99) across all combinations of noise amplitude σ and Lévy index α that can be considered as weak noise levels.This suggests that within the employed limits both parameters do not strongly affect the performance of the CSD indicators.The depicted early warning pipeline entails the usage of the interquartile range as an early warning indicator and the Kendall-τ to assess an increasing slope.In addition, the underlying time series data was generated by our model modification which resolves small timescales.
In cases that small timescales are not resolved, we obtain a low average recall of 0.11 (fig.3b).Moreover, the recall is low across all admissible parameter combinations.Thus, also for Gaussian noise (α = 2), the early warning does not work, which underlines the necessity to resolve small timescales.In other terms, if this were the case for the real world, i.e. the rainforest would only respond very slowly to perturbations, a declining forest resilience would not be measurable with CSD indicators.For such a transient situation, even considering weak Gaussian noise does not allow us to measure forest resilience solely on the basis of CSD indicators.Hence, in the following, we will only consider modelled time series, where a response to disturbances on small timescales is resolved.
If critical slowing down is assessed on a filtered time series, we find a low false  positive rate of on average 0.02 for the non-parametric Kendall-τ slope on IQR time series (fig.4a).Here, the filtering is leveraged as a post-hoc adjustment to assess the significance of a decline: by random chance, one might detect a significant slope even if the magnitude of the autocorrelation is very low.Hence, we flag those early warnings, where the temporal autocorrelation is below 0.5.We refer to the resulting indicator as Adjusted Kendall-Tau Slope.Note that the autocorrelation is particularly suitable to perform this filtering because there is a natural way of thresholding, in contrast to the standard deviation or the IQR.When including such filtering, the early warning (forest resilience) pipeline produces few false alarms.
In constrast, when early warning is detected on raw (unfiltered) time series, we detect a non-neglibile average false positive rate of 0.10, again for the case of non-parametric Kendall-τ slope on IQR time series (fig.4b).Hence, the early warning pipeline without the filtering of perturbations according to their temporal autocorrelation, raises false alarms.For an ecological system, this means a declining resilience measured solely by a CSD indicator should always be double-checked, for instance against the magnitude of the temporal autocorrelation in historical time series.Otherwise, a false alarm may be raised.
Table 1 summarizes the key findings by comparing mean recall and mean false positive rates across a selection of scenarios.First, using Kendall-tau slopes in our model (including small time scales), high recall can be achieved (sixth row).Second, adjusting the early warning for the magnitude of the autocorrelation, low false-positive rates at high recall are possible (last row).For the original van Nes et al. [37] model, recall is always low, irrespective of the processing (exemplified in the first two rows, but consistent among other scenarios, not displayed in Table 1).For our model, only the Kendall-tau slope gives consistently high recall, linear and Theil-Sen slopes suffer from jumps in the noise structure (second block of rows).The type of indicator used to assess CSD (i.e.rainforest resilience) is less relevant, we find the interquartile range (IQR) slightly outperforms standard deviation and autocorrelation, but all three measures are valid options (third block of rows).The average false positive rate before adjusting is around 10%, and can be reduced to only 2% by removing false positives with low autocorrelation.

Discussion
Our work suggests that critical slowing down can be used to assess resilience loss due to an approaching critical transition also in the presence of extreme events modelled by α-stable Lévy noise.In the case of the Amazon rainforest, this confirms the empirical results by other studies indicating a destabilization of the system [5,7,8,53,54] as being more generally applicable than previously thought.
The particular tropical rainforest model used in this study is a strongly simplified rainforest model.However, it can be understood as a special case of a very general model: a double-well potential, the mathematical normal form of a saddle-node bifurcation.For transitions in a double-well potential, our findings hold and are robust (see Appendix A).Hence, they are not limited to just the particular choice of the conceptual model in this study but are more widely applicable to any system exhibiting such critical transitions.For instance, many tipping elements of the Earth system have been modelled through special cases of the double-well potential [4,55,56].Also some other models of tropical rainforests [38] belong to the double-well potential family.Furthermore, we claim our analysis is relevant for the actual rainforest system, which is highly complex.To support such statement, one may consider increasing complexity of the model, e.g. by studying global dynamical vegetation models.Due to their high-dimensionality, they usually cannot be easily represented by a potential landscape.However, for CMIP6 models, abrupt local forest dieback has been diagnosed in the Amazon basin [57].Hence, the existence of critical transitions is indicated and locally such a transition can again be reasonably well approximated by simple double-well potential models, as done in this study.
Not all non-Gaussian disturbances necessarily are of α-stable Lévy-type.Instead, in some circumstances, coloured noise has been observed in ecological systems and in particular in tropical rainforest [58][59][60][61][62].In such cases, the two most commonly postulated colour noises are pink noise and red noise.For red noise, which displays an auto-correlated noise structure, resilience measures based on CSD indicators need to be adapted, but then they work as in the case of white noise [33].Pink noise has power-law tails but does not lead to jumps in the forest state evolution.Therefore, we find that our results for α-stable Lévy noise are robust for pink noise (see Appendix B and Table B1 for details).Hence, for a wide range of non-Gaussian noise observed in ecology, CSD indicators are valid choices to measure tropical rainforest resilience.
Two further points regarding the accuracy of our mathematical model require caution: First, we have chosen to model forest disturbances via α-stable Lévy noise to honour the prevalence of extreme events in observations.However, as the variable representing the state of the rainforest is bounded between 0 and 100% (tree cover density), the far end of the noise tail needs to be disregarded, because tree cover cannot fall below 0% nor go beyond 100%.This restricts the conceptual modelling capabilities of the α-stable noise model.It does however not pose a practical problem for interpreting the model data as observations from a multi-stable forest system because an exceedingly extreme disturbance always simply leads to a tipped system.Our work demonstrates that the concept of CSD holds in the case of α-stable noise distributions when observing the time span before tipping.Second, due to ongoing deforestation for many decades, one might argue the Amazon is not in a steady state, but rather on a transient.Hence, the equilibrium assumption of critical slowing down, whereby a slow change in the external forcing only changes the equilibrium state, but does not keep the system in disequilibrium for long, may be violated.Nevertheless, studies have found that the Amazon rainforest may approach a tipping point due to deforestation [63,64].

Conclusion
We find robustness of critical slowing down indicators used to measure resilience in multi-stable systems driven by α-stable Lévy noise.The presence of disturbances with power-law spectrum occurring in single discrete jumps does not affect the identification of early warning of a critical transition as long as the jump size is small enough to avoid immediate, noise-induced transitions between alternative stable states.For this purpose, we find it is ideal to test the interquartile range with Kendall-Tau slope for significant increases and filter for cases with low autocorrelation.With such processing, high recall and low false positive rates can be achieved.Most recent work computing resilience indicators based on remote sensing in the Amazon basin follow a similar procedure [7,26].In particular, they assess rainforest resilience with critical slowing down indicators and find a resilience decline.Our work emphasizes that such empirical findings are not corroborated in the presence of non-Gaussian disturbances, which could not be ruled out previously.Hence, it adds to the increasing evidence that the Amazon rainforest's resilience has been declining in recent decades, irrespective of the actual nature of the noise (white, coloured, or Lévy noise).Future work may extend the analysis to include a spatial component.Spatial processes are highly relevant for tropical vegetation health and resulting spatial patterns [35,36], e.g.patchiness, have been introduced as another type of early warning signal (i.e.resilience indicator) [65].

Appendix A. Double-well potential
We repeat our α-stable Lévy noise experiments with a generic model based on a doublewell potential.More specifically, we construct a double-well potential with similar time scales and value range as the Amazon model with small time scales introduced in the main body of this work.Let x ∈ [0, 100] be an abstraction of the rainforest state and c ∈ R be an external forcing (think climate change).Our model becomes the first-order ODE: With a time scale τ = 1000, a value range a = 20 and a value shift b = −3 as fixed parameters.For c < −1, Eq.A.1 has one stable fixed point (the rainforest state, x ≈ 80), for −1 ≤ c ≤ 1, there are two stable fixed points exist and for c > 1 there is again only one stable fixed point exists (the savannah state x ≈ 40).As in the main text, we choose α-stable Lévy noise for the noise term dN t .These parameter settings allow us to perform simulations in the same set-up as described in Fig. 1, with the minor difference that we simulate climate change by linearly increasing the forcing from c = −1.5 to c = 1.5.The results are consistent with those in the main body.Fig. A1 shows the recall across a variety of noise settings, for two cases: 1. adjusted Kendall-Tau slopes on the interquartile range, where recall is always high, and 2. linear slopes on the standard deviation, where some deterioration in recall can be observed towards the strong noise regime.The adjustment for a low false positive rate is less effective in the case of the double-well potential: without it, Kendall-Tau slopes on IQR get 9.3% false positives, while with the adjustment, the FPR drops to 8.6%.Likely this is due to the threshold of AC(1) = 0.5 introduced in the main body, which seems valid for the Amazon rainforest model, but not for the double-well potential.= (1 + λ∆t) α c α x (∆t) + σ α ∆t ⇔ c α x (∆t) = σ α ∆t 1 − (1 + λ∆t) α In the limit of ∆t → 0, this implies that This result is consistent with the known variance c 2 x = −σ 2 2λ for the Gaussian white noise case of α = 2.
Since the scaling parameter c x dictates the distribution width of the observed process, we may posit a direct influence of the linear restoring rate λ on the observed variance and the IQR, analogous to the case of Gaussian white noise.We have to bear in mind however, that in this setting of an unbounded observable, the variance of the α-stable distribution is ill-defined and will numerically diverge to infinity when applying law of large numbers estimators.In the application of a bounded forest state variable, we expect the finite variance to be an increasing function of λ, which can thus function as a CSD indicator.Further analysis is needed to theoretically motivate the use of AC(1) in a similar fashion.However, as has been laid out during the analysis within the main text, its use is warranted on a numerical basis.

Figure 1 .
Figure 1.Overview of the study design.a) Tropical rainforest model bifurcation diagram:Depending on the precipitation level, three different equilibrium states may exist, namely rainforest, savannah and desert.b) Many natural stressors can be modeled through α-stable Lévy noise, i.e. in contrast to Gaussian noise (α = 2), the tails have power-law decay and hence large events can happen.c) We simulate a climate change scenario, during which we force the precipitation level to decrease below the critical threshold of the rainforest-savannah transition.We compute the rolling autocorrelation AC(1) before the critical transition and observe increases near single large disturbance events and in approach to the critical transition.d) The critical slowing down indicator time series (in this case: AC(1)) gets converted into a binary early warning indicator.For this, first a slope is calculated in a window approaching the critical transition.Then, the statistical significance of this slope is assessed by comparing to a null distribution of random phase shift surrogates with a one-sided test.If the slope is significant, there is early warning (i.e.decreasing rainforest resilience).

Figure 2 .
Figure 2. a) The potential landscape of the tropical vegetation model used in this study, for varying precipitation levels (colors).The dotted line represents the original potential used in[37].The solid line includes the modifications to represent small time scales introduced in this work.b) The return time it takes for the forest to recover from a disturbance.In the original van Nes et al.[37] model (dash-dotted), return times even for very small disturbances are in the hundreds of years.Our work adds small time scales (solid), i.e. recovery from small disturbances e.g.due to synopticscale weather variations is quick.

RecallFigure 3 .
Figure 3. Recall across various noise amplitudes and anomaly indices (α = 2 is Gaussian noise).Shown are only those noise configurations that can be considered as weak noise determined by the observed first passage time.Panel a) shows a processing chain that leads to high recall, i.e. the power-law disturbances are properly dealt with.Recall is close to 1.0 for all noise configurations, with slight decrease close to the strong noise regime.Panel b) in contrast shows no proper treatment of power-law disturbances, which leads to low recall close to 10% ± 2%.

Figure 4 .
Figure 4. False positive rate (FPR) for parameter ranges also shown in Fig. 3. Panel a) displays a processing chain that leads to low FPR, i.e. the power-law disturbances are properly dealt with.FPR is close to 0.0 for noise configurations far from the strong noise regime, but increases close to the strong noise regime.Panel b) in contrastshows the effect of improper treatment of power-law disturbances.This leads to a false positive rate close to 10%, but with a higher variation across noise configurations.

Figure A1 .
Figure A1.Same as fig.3, but using a double-well potential as the model.Recall across various noise amplitudes and anomaly indices (α = 2 is Gaussian noise).Shown are only those noise configurations that can be considered as weak noise determined by the observed first passage time.Panel a) shows a processing chain that leads to high recall, i.e. the power-law disturbances are properly dealt with.Recall is close to 1.0 for all noise configurations, with a slight decrease close to the strong noise regime.Panel b) in contrast shows no proper treatment of power-law disturbances, which leads to reduced recall closer to the strong noise regime.

P < 0.05 ? Yes No Steady State Climate Change New Normal
model of a tropical vegetation ecosystem for simulating regime shifts.In this work, we investigate tropical rainforest resilience under power-law noise.Forest resilience is frequently measured with indicators exploiting critical slowing down

Table 1 .
[37]age recall and false positive rate (FPR) for various processing chains.Averages are computed across a wide choice of noise amplitudes and anomaly indices, as long as the resulting noise can be considered weak (as measured by the first passage time).The first two rows show results for the original van Nes et al.[37]model, for which critical slowing down does not work due to a lack of representation of small time scales.The second block of rows presents different choices for computing the slope of the early warning indicator when considering our model, which includes small time scales: the non-linear parametric Kendall-Tau slope is robust and thus preferrential.
).Hence, early warning works with few misses.This corresponds to a measure of forest resilience